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# Identifying Groups of Strongly Correlated Variables through Smoothed Ordered Weighted $L_1$-norms

*Proceedings of the 20th International Conference on Artificial Intelligence and Statistics*, PMLR 54:1123-1131, 2017.

#### Abstract

The failure of LASSO to identify groups of correlated predictors in linear regression has sparked significant research interest. Recently, various norms were proposed, which can be best described as instances of ordered weighted $\ell_1$ norms (OWL), as an alternative to $\ell_1$ regularization used in LASSO. OWL can identify groups of correlated variables but it forces the model to be constant within a group. This artifact induces unnecessary bias in the model estimation. In this paper we take a submodular perspective and show that OWL can be posed as the Lovász extension of a suitably defined submodular function. The submodular perspective not only explains the group-wise constant behavior of OWL, but also suggests alternatives. The main contribution of this paper is smoothed OWL (SOWL), a new family of norms, which not only identifies the groups but also allows the model to be flexible inside a group. We establish several algorithmic and theoretical properties of SOWL including group identification and model consistency. We also provide algorithmic tools to compute the SOWL norm and its proximal operator, whose computational complexity $O(d\log d)$ is significantly better than that of general purpose solvers in $O(d^2\log d)$. In our experiments, SOWL compares favorably with respect to OWL in the regimes of interest.